Commit 8d66b8b4 by Jim Hefferon

### some elimination of passive voice

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 ... ... @@ -8,8 +8,9 @@ These two examples from high school science \cite{Onan} give a sense of how they arise. The first example is from Physics.\index{Physics problem} Suppose that we are given three objects, one with a mass known to be 2~kg, and are asked to find the unknown masses. Suppose that we have three objects, one with a mass known to be 2~kg. We are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances. \begin{center} ... ... @@ -19,11 +20,10 @@ experimentation with a meter stick produces these two balances. \end{center} We know that the moment of each object is its mass times its distance from the balance point. We also know that for balance we must have that the sum of moments on the left equals the sum of moments on We also know that for the masses to balance we must have that the sum of moments on the left equals the sum of moments on the right. That gives a system of two equations. That gives us a system of two equations. \begin{align*} 40h+15c &= 100 \\ 25c &= 50+50h ... ... @@ -37,7 +37,7 @@ trinitrotoluene $\hbox{C}_7\hbox{H}_5\hbox{O}_6\hbox{N}_3$ along with the byproduct water (conditions have to be controlled very well\Dash trinitrotoluene is better known as TNT). In what proportion should we mix those components? In what proportion should we mix them? The number of atoms of each element present before the reaction \begin{equation*} x\,{\rm C}_7{\rm H}_8\ +\ y\,{\rm H}{\rm N}{\rm O}_3 ... ... @@ -45,7 +45,7 @@ The number of atoms of each element present before the reaction z\,{\rm C}_7{\rm H}_5{\rm O}_6{\rm N}_3\ +\ w\,{\rm H}_2{\rm O} \tag*{}\end{equation*} must equal the number present afterward. Applying that to the elements C, H, N, and O in turn gives Applying that in turn to the elements C, H, N, and O gives this system. \begin{align*} 7x &= 7z \\ ... ... @@ -75,7 +75,7 @@ This chapter shows how to solve any such system. \subsection{Gauss' Method} \begin{definition} A \definend{linear combination}\index{linear combination} of $$x_1,x_2,\ldots,x_n$$ has the form $$x_1$$, \ldots, $$x_n$$ has the form \begin{equation*} a_1x_1+a_2x_2+a_3x_3+\cdots+a_nx_n \end{equation*} ... ... @@ -89,7 +89,8 @@ where An $$n$$-tuple $$(s_1,s_2,\ldots ,s_n)\in\Re^n$$ is a \definend{solution}\index{linear equation!solution of} % of, or \definend{satisfies}, that equation if substituting the numbers $s_1$, \ldots, $s_n$ for the variables gives a true statement: $s_1$, \ldots, $s_n$ for the variables $x_1$, \ldots, $x_n$ gives a true statement: $a_1s_1+a_2s_2+\ldots+a_ns_n=d$. A \definend{system of linear equations}\index{linear equation!system of}% ... ... @@ -127,9 +128,10 @@ In contrast, $$(5,-1)$$ is not a solution. Finding the set of all solutions is \definend{solving}\index{system of linear equations!solving} the system. No guesswork or good fortune is needed to solve a linear system. There is an algorithm that always works. The next example introduces that algorithm, called We don't need guesswork or good luck, there is an algorithm that always works. This algorithm is called \definend{Gauss' method}\index{Gauss' method}% \index{system of linear equations!Gauss' method} (or \definend{Gaussian elimination}\index{Gaussian elimination}% ... ... @@ -138,10 +140,10 @@ or \definend{linear elimination}\index{linear elimination}% \index{system of linear equations!linear elimination}% \index{system of linear equations!elimination}% \index{elimination}). It transforms the system, step by step, into one with a form that is easily solved. We will first illustrate how it goes and then we will see the formal statement. % It transforms the system, step by step, into one % with a form that we can easily solve. % We will first illustrate how it goes and then we will see the % formal statement. \begin{example} To solve this system ... ... @@ -152,11 +154,11 @@ To solve this system \frac{1}{3}x_1 &+ &2x_2 & & &= &3 \end{linsys} \end{equation*} we repeatedly transform it until it is in a form that is easy to solve. Below there are three transformations. we transform it, step by step, until it is in a form that we can easily solve. The first is to rewrite the system by interchanging the first and third row. The first transformation is to rewrite the system by interchanging the first and third row. \begin{eqnarray*} \quad &\grstep{ \text{swap row 1 with row 3} } ... ... @@ -194,16 +196,17 @@ a form where we can easily find the value of each variable. The bottom equation shows that $$x_3=3$$. Substituting $3$ for $$x_3$$ in the middle equation shows that $$x_2=1$$. Substituting those two into the top equation gives that $$x_1=3$$ and so the system has a unique solution: gives that $$x_1=3$$. Thus the system has a unique solution; the solution set is $\{\,(3,1,3)\,\}$. \end{example} Most of this subsection and the next one consists of examples of solving linear systems by Gauss' method. We will use it throughout this book. We will use it throughout the book. It is fast and easy. But before we get to those examples, we will first show that But before we do those examples we will first show that this method is also safe in that it never loses solutions or picks up extraneous solutions. ... ... @@ -223,7 +226,7 @@ then the two systems have the same set of solutions. \end{theorem} Each of those three operations has a restriction. Multiplying a row by $$0$$ is not allowed because that Multiplying a row by $$0$$ is not allowed because obviously that can change the solution set of the system. Similarly, adding a multiple of a row to itself is not allowed because adding $$-1$$ times the row to itself has the effect of multiplying the row ... ... @@ -260,16 +263,17 @@ Consider this swap of row~$i$ with row~$j$. \end{equation*} }% end change of arraystretch The $$n$$-tuple $$(s_1,\ldots\,,s_n)$$ satisfies the system before the swap if and only if substituting the values, the $s$'s, for the variables, the $x$'s, gives true statements: if and only if substituting the values for the variables, , the $s$'s for the $x$'s, gives true statements: $a_{1,1}s_1+a_{1,2}s_2+\cdots+a_{1,n}s_n=d_1$ and \ldots\ $a_{i,1}s_1+a_{i,2}s_2+\cdots+a_{i,n}s_n=d_i$ and \ldots\ $a_{j,1}s_1+a_{j,2}s_2+\cdots+a_{j,n}s_n=d_j$ and \ldots\ $a_{m,1}s_1+a_{m,2}s_2+\cdots+a_{m,n}s_n=d_m$. In a requirement consisting of statements joined with and' we can rearrange the order of the statements, so that In a sentence consisting of statements joined with and' we can rearrange the order of the statements. Thus this requirement is met if and only if $a_{1,1}s_1+a_{1,2}s_2+\cdots+a_{1,n}s_n=d_1$ and \ldots\ $a_{j,1}s_1+a_{j,2}s_2+\cdots+a_{j,n}s_n=d_j$ ... ...
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 ... ... @@ -261,7 +261,6 @@ %------------------------- code listings \usepackage{listings} \lstset{basicstyle=\small\ttfamily, language=python, commentstyle=\textit, keywordstyle=\color{blue}\bfseries, showstringspaces=false, ... ...
 \thispagestyle{plain} \setlength{\parskip}{.7ex} \bigskip \vspace*{1.35in} \vspace*{1.25in} \noindent{\Huge\bf Preface} \vspace*{.4in} % \bigskip ... ... @@ -23,7 +23,7 @@ using many examples as well as extensive and careful exercises. The developmental approach is what most recommends this book so I will elaborate. Courses at the beginning of most mathematics programs The courses at the beginning of a mathematics program focus less on theory and more on calculating. Later courses ask for mathematical maturity:~the ability to follow different ... ... @@ -31,8 +31,10 @@ types of arguments, a familiarity with the themes that underlie many mathematical investigations such as elementary set and function facts, and a capacity for some independent reading and thinking. Linear algebra is an ideal spot to work on the transition. and a capacity for some independent reading and thinking. Some programs have a separate course devoted to developing maturity and some do not. In either case, linear algebra is an ideal spot to work on this transition. It comes early in a program so that progress made here pays off later, but also comes late enough that students are serious about mathematics. ... ... @@ -43,14 +45,14 @@ contradiction, and proofs by induction. And, examples are plentiful. Helping readers start the transition to being serious students of the subject of mathematics itself means taking the mathematics seriously, so mathematics requires taking the mathematics seriously, so all of the results here are proved. On the other hand, we cannot assume that students have already arrived and so in contrast with more abstract texts, we give many examples, in contrast with more abstract texts this book is filled with examples, often quite detailed. Some linear algebra books ... ... @@ -58,47 +60,41 @@ begin with extensive computations of linear systems, matrix multiplications, and determinants. Then, when vector spaces and linear maps finally appear, vector spaces and linear maps finally appear and definitions and proofs start, the abrupt change can bring students to an abrupt stop. In this book, while we start with While this book starts with a computational topic, linear reduction, from the first we do more than compute. We do linear systems quickly but completely, including the proofs needed to justify what we are computing. The linear reduction chapter includes the proofs needed to justify what we are computing. Then, with the linear systems work as motivation and at a point where the study of linear combinations seems natural, so that the study of linear combinations seems natural, the second chapter starts with the definition of a real vector space. In the schedule below, this occurs by the end of the third week. Another example of the emphasis here on motivation and naturalness Another example of this book's emphasis on motivation and naturalness is that the third chapter on linear maps does not begin with the definition of homomorphism, but with isomorphism. Isomorphism is natural: students themselves does not begin with the definition of homomorphism. Rather, we start with the definition of isomorphism, which is natural: students themselves observe that some spaces are just like'' others. After that, the next section takes the reasonable step of isolating the operation-preservation idea to define homomorphism. This approach loses mathematical slickness over the reversed order, This approach loses mathematical slickness, but it is a good trade because it gives to students a large gain in sensibility. %In short: one aim of our developmental approach is %that students should see how the ideas arise, and should be able to %picture themselves doing the same type of work. % The developmental approach most clearly come out in the % exercises. A student progresses in mathematics most while doing exercises, so the ones here have been selected with great care. A student progresses most in mathematics while doing exercises so the ones here have gotten close attention. Problem sets start with simple checks and range up to reasonably involved proofs. Since instructors usually assign about a dozen exercises I have tried to put about two dozen in each set, thereby providing a selection. There are even a few problems that are puzzles thereby giving a selection. There are even a few that are puzzles taken from various journals, competitions, or problems collections. These are marked with a ... ... @@ -125,14 +121,14 @@ Applications and computing are interesting and vital aspects of the subject. Consequently, each of this book's chapters closes with a few topics in those areas. They are brief enough that an instructor can do one in a day's class or can assign them as independent or small-group projects. Most simply give a reader a taste of the subject, discuss how linear algebra comes in, point to some further reading, and give a few exercises. They are brief enough that an instructor can do one in a day's class, or can assign them as independent or small-group projects. With a few of these, readers can see for themselves that linear algebra is a tool Whether they figure formally in the course or not, these help readers see for themselves that linear algebra is a tool that a professional must have. ... ... @@ -155,23 +151,23 @@ See the contact information on that page. \newcommand{\classday}[1]{\textsc{#1}} \newcommand{\colwidth}{1.35in} \newcommand{\colwidth}{1.25in} %\vspace*{.5in} \medskip \noindent{\bf If you are reading this book on your own.} %\smallskip % The emphasis here on motivation and development make this text popular for self-study. This book's emphasis on motivation and development, and its availibility, make it widely used for self-study. If you are an independent student then you may find some advice helpful. In particular, while an experienced instructor knows what subjects and pace suit their class, you may find useful these couple of timetables for a semester. pace suit their class, you may find useful these timetables for a semester. The first focuses on core material. \begin{center} \begin{center} \small \begin{tabular}{r|*{2}{p{\colwidth}}l} \textit{week} &\textit{Monday} ... ... @@ -195,7 +191,7 @@ The first focuses on core material. \end{center} The second is more ambitious. It supposes that you know section One.II, the elements of vectors. \begin{center} \begin{center} \small \begin{tabular}{r|*{2}{p{\colwidth}}l} \textit{week} &\textit{Monday} ... ... @@ -233,7 +229,7 @@ I have marked a good sample with \recommendationmark's in the margin. For all of them, you must justify your answer either with a computation or with a proof. Be aware that few inexperienced people can write correct proofs; try to find a trained person to work with you on these. try to find a knowledgable person to work with you on these. \bigskip Finally, a caution for all students, independent or not:~I ... ...
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